Scattering for NLS with a sum of two repulsive potentials
Annales de l'Institut Fourier, Volume 70 (2020) no. 5, pp. 1847-1869.

We prove the scattering for a defocusing nonlinear Schrödinger equation with a sum of two repulsive potentials with strictly convex level surfaces, thus providing a scattering result in a trapped setting similar to the exterior of two strictly convex obstacles.

Nous montrons la diffusion pour une équation de Schrödinger non linéaire défocalisante avec une somme de deux potentiels répulsifs dont les surfaces de niveau sont strictement convexes. Il s’agit d’un résultat dans une géométrie captante similaire à l’extérieur de deux obstacles strictement convexes.

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DOI: 10.5802/aif.3385
Classification: 35Q55, 35B40
Keywords: nonlinear Schrödinger equation, scattering, trapped trajectories, Morawetz estimates, concentration-compactness/rigidity
Mot clés : équation de Schrödinger non linéaire, diffusion, trajectoires captées, estimées de Morawetz, concentration-compacité/rigidité
Lafontaine, David 1

1 University of Bath Dept. of Mathematical Sciences Bath BA2 7AY (United Kingdom)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Lafontaine, David. Scattering for NLS with a sum of two repulsive potentials. Annales de l'Institut Fourier, Volume 70 (2020) no. 5, pp. 1847-1869. doi : 10.5802/aif.3385. https://aif.centre-mersenne.org/articles/10.5802/aif.3385/

[1] Banica, Valeria; Visciglia, Nicola Scattering for NLS with a delta potential, J. Differ. Equations, Volume 260 (2016) no. 5, pp. 4410-4439 | DOI | MR

[2] Carles, Rémi On semi-classical limit of nonlinear quantum scattering, Ann. Sci. Éc. Norm. Supér., Volume 49 (2016) no. 3, pp. 711-756 | DOI | MR

[3] Cazenave, Thierry; Weissler, Fred B. Rapidly decaying solutions of the nonlinear Schrödinger equation, Commun. Math. Phys., Volume 147 (1992) no. 1, pp. 75-100 | DOI | MR | Zbl

[4] Forcella, Luigi; Visciglia, Nicola Double scattering channels for 1D NLS in the energy space and its generalization to higher dimensions, J. Differ. Equations, Volume 264 (2018) no. 2, pp. 929-958 | DOI | MR | Zbl

[5] Foschi, Damiano Inhomogeneous Strichartz estimates, J. Hyperbolic Differ. Equ., Volume 2 (2005) no. 1, pp. 1-24 | DOI | MR | Zbl

[6] Goldberg, Michael; Schlag, Wilhelm Dispersive estimates for Schrödinger operators in dimensions one and three, Commun. Math. Phys., Volume 251 (2004) no. 1, pp. 157-178 | DOI | MR | Zbl

[7] Hong, Younghun Scattering for a nonlinear Schrödinger equation with a potential, Commun. Pure Appl. Anal., Volume 15 (2016) no. 5, pp. 1571-1601 | DOI | MR | Zbl

[8] Ivanovici, Oana; Planchon, Fabrice On the energy critical Schrödinger equation in 3D non-trapping domains, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 27 (2010) no. 5, pp. 1153-1177 | DOI | Numdam | MR

[9] Keel, Markus; Tao, Terence Endpoint Strichartz estimates, Am. J. Math., Volume 120 (1998) no. 5, pp. 955-980 | DOI | MR

[10] Kenig, Carlos E.; Merle, Frank Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., Volume 166 (2006) no. 3, pp. 645-675 | DOI | MR

[11] Lafontaine, David Scattering for NLS with a potential on the line, Asymptotic Anal., Volume 100 (2016) no. 1-2, pp. 21-39 | DOI | MR

[12] Lafontaine, David About the wave equation outside two strictly convex obstacles (2017) (https://arxiv.org/abs/1711.09734)

[13] Lafontaine, David; Laurent, C. Scattering for critical nonlinear waves outside some strictly convex obstacles (Work in progress)

[14] Nakanishi, Kenji Energy scattering for nonlinear Klein–Gordon and Schrödinger equations in spatial dimensions 1 and 2, J. Funct. Anal., Volume 169 (1999) no. 1, pp. 201-225 | DOI | MR

[15] Planchon, Fabrice; Vega, Luis Bilinear virial identities and applications, Ann. Sci. Éc. Norm. Supér., Volume 42 (2009) no. 2, pp. 261-290 | DOI | Numdam | MR

[16] Planchon, Fabrice; Vega, Luis Scattering for solutions of NLS in the exterior of a 2D star-shaped obstacle, Math. Res. Lett., Volume 19 (2012) no. 4, pp. 887-897 | DOI | MR

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